Trigonometry Calculator | Sin, Cos, Tan
Calculate sine, cosine, tangent, and their inverses for any angle.
Angle Unit
Special Angles
What Is the Trigonometry Calculator | Sin, Cos, Tan?
Trigonometry deals with the relationships between angles and the ratios of sides in triangles. The six trigonometric functions, sine, cosine, tangent, and their reciprocals, are foundational to physics, engineering, signal processing, and computer graphics.
Angle Unit Systems
- ›Degrees (°): circle divided into 360 parts, everyday geometry and navigation
- ›Radians (rad): arc length / radius; a full circle = 2π rad ≈ 6.2832, calculus standard
- ›Gradians (grad): circle divided into 400 parts; 90° = 100 grad, surveying in France/Germany
- ›Conversion: deg × π/180 = rad; deg × 10/9 = grad
Key Identities
- ›Pythagorean: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ
- ›Reciprocal: csc = 1/sin; sec = 1/cos; cot = 1/tan
- ›Even/Odd: cos(−θ) = cos θ (even); sin(−θ) = −sin θ (odd)
- ›tan θ is undefined where cos θ = 0 (at 90°, 270°, and all odd multiples of 90°)
Formula
sin θ
Opposite / Hypotenuse
cos θ
Adjacent / Hypotenuse
tan θ
sin θ / cos θ = Opposite / Adjacent
csc θ
1 / sin θ
sec θ
1 / cos θ
cot θ
1 / tan θ = cos θ / sin θ
Pythagorean Identity
sin²θ + cos²θ = 1
Angle Conversion
rad = deg × π/180 = grad × π/200
How to Use
- 1Select the angle unit: Degrees, Radians, or Gradians.
- 2Click a special angle button (0°, 30°, 45°, 60°, 90°...) for instant exact values, or type any angle.
- 3Results update live as you type, no need to click Calculate each time.
- 4The Trig Functions panel shows all 6 functions; "Undefined" (in red) flags division by zero.
- 5Click "copy" next to any value to copy it to the clipboard.
- 6Inverse Functions shows arcsin(sin θ), arccos(cos θ), arctan(tan θ), all in degrees.
- 7Identity Verification confirms sin²θ + cos²θ = 1 to 10 decimal places.
Example Calculation
Compute all six trig functions for θ = 30°:
θ = 30° = π/6 rad = 33.3333 grad
sin(30°) = 0.5 csc(30°) = 2
cos(30°) = 0.86602540 sec(30°) = 1.15470054
tan(30°) = 0.57735027 cot(30°) = 1.73205081
Identity check:
(0.5)² + (0.8660)² = 0.25 + 0.75 = 1.0000000000 ✓
Inverse functions:
arcsin(0.5) = 30°
arccos(0.8660) = 30°
arctan(0.5774) = 30°
Key Insight
30° is one of the exact special angles. Note that sin(30°) = cos(60°) = 0.5, and tan(30°) = 1/√3 ≈ 0.5774. These exact values derive from the 30-60-90 right triangle with sides 1 : √3 : 2.
Understanding Trigonometry | Sin, Cos, Tan
Special Angles, Exact Values Reference
| Angle (°) | Radians | sin θ | cos θ | tan θ | Notes |
|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | Origin |
| 30° | π/6 | 0.5000 | 0.8660 | 0.5774 | 30-60-90 triangle |
| 45° | π/4 | 0.7071 | 0.7071 | 1 | 45-45-90 triangle |
| 60° | π/3 | 0.8660 | 0.5000 | 1.7321 | 30-60-90 triangle |
| 90° | π/2 | 1 | 0 | Undefined | Max sine |
| 120° | 2π/3 | 0.8660 | −0.5000 | −1.7321 | Second quadrant |
| 135° | 3π/4 | 0.7071 | −0.7071 | −1 | Second quadrant |
| 150° | 5π/6 | 0.5000 | −0.8660 | −0.5774 | Second quadrant |
| 180° | π | 0 | −1 | 0 | Negative x-axis |
| 270° | 3π/2 | −1 | 0 | Undefined | Min sine |
| 360° | 2π | 0 | 1 | 0 | Full circle |
Frequently Asked Questions
What are the six trigonometric functions?
- ›Primary: sin θ = opp/hyp; cos θ = adj/hyp; tan θ = opp/adj = sin/cos
- ›Reciprocal: csc θ = 1/sin θ (undefined at 0°, 180°)
- ›Reciprocal: sec θ = 1/cos θ (undefined at 90°, 270°)
- ›Reciprocal: cot θ = 1/tan θ = cos/sin (undefined at 0°, 180°)
What is the difference between degrees and radians?
- ›Degrees: 360° = full circle; convenient for everyday geometry, navigation, maps
- ›Radians: 2π rad = full circle; 1 rad = the angle where arc length = radius
- ›Only in radians is d/dx(sin x) = cos x, in degrees you get an extra π/180 factor
- ›Most programming languages (including JavaScript) use radians for Math.sin, Math.cos
Why is tan(90°) undefined?
- ›tan θ = sin θ / cos θ; at 90°, cos 90° = 0 → division by zero
- ›The tangent function approaches ±∞ as θ → 90° from each side
- ›Also undefined at 270°, ±450°, etc., all odd multiples of 90°
- ›csc is undefined where sin = 0 (at 0°, 180°); sec is undefined where cos = 0
What are the exact values for special angles?
- ›sin(0°)=0, sin(30°)=½, sin(45°)=√2/2, sin(60°)=√3/2, sin(90°)=1
- ›cos is the same sequence in reverse: cos(0°)=1 down to cos(90°)=0
- ›30-60-90 triangle has sides 1 : √3 : 2, source of the 30° and 60° values
- ›45-45-90 triangle has sides 1 : 1 : √2, source of sin(45°) = cos(45°) = √2/2
What is the Pythagorean identity?
- ›sin²θ + cos²θ = 1, holds for every angle θ, including complex numbers
- ›Divide by cos²θ → 1 + tan²θ = sec²θ
- ›Divide by sin²θ → cot²θ + 1 = csc²θ
- ›On the unit circle: sin θ = y, cos θ = x, and x² + y² = 1 (the circle equation), proof is immediate
What are gradians and who uses them?
- ›400 gradians = full circle; 100 gradians = right angle (convenient for surveyors)
- ›Also called "gon" or "grade" in engineering contexts
- ›Used in civil engineering surveys in France, Germany, Sweden, and parts of Eastern Europe
- ›Conversion: deg × 10/9 = grad; 45° = 50 grad; 90° = 100 grad; 180° = 200 grad
What do the inverse trig functions return?
- ›arcsin: principal range [−90°, 90°]; arcsin(sin 150°) = 30° (not 150°)
- ›arccos: principal range [0°, 180°]; arccos(cos 200°) = 160°
- ›arctan: principal range (−90°, 90°); arctan(tan 200°) = 20°
- ›The calculator shows the principal value, to recover the original angle, use quadrant analysis